Theoretical astrophysicists have been developing radiative transfer theory for a long time. However, they have been primarily concerned with stellar atmospheres, within which the scattering of light is isotropic. In the atmospheres of the planets, light scattering by an elemen­

tary volume is anisotropic. This fact severely complicates the theory. Nevertheless, in recent years the theory of radiative transfer for anisotropic scattering has made considerable pro­ gress and has been increasingly used in the study of planetary atmospheres. The present monograph has been written for the purpose of summarizing the results of work in this area.

The monograph is concerned mainly with the theory of radiative transfer for anisotropic scattering. The first eight chapters deal with the general problem of multiple scattering of light in an atmosphere consisting of plane-parallel layers illuminated by parallel radiation.

In the following two chapters, the theory is applied to the determination of the physical characteristics of planetary atmospheres. The last chapter discusses the theory of radiative transfer in spherical atmospheres, which is necessary for the interpretation of observations made from spacecraft.

The emphasis in the monograph on the theory rather than its application is easily under­ stood; the theory is designed not only for the interpretation of existing observational data, but also for that to be gathered in the future. One must also bear in mind that the theory of radiative transfer is utilized in related sciences, such as meteorology and oceanology, and also in certain branches of physics and chemistry.

Chapter 1 Basic Equations 1

1.1 The scattering of light by an elementary volume 1

1.2 The equation of radiative transfer 5

1.3 The Basic Problem 8

1.4 Integral equations for the Source Function 12

1.5 The diffuse radiation field 15

1.6 The case of pure scattering 19

1.7 Methods for solving the problem 21

Chapter 2 Semi-infinite Atmospheres 24

2.1 The Radiation field in Deep Layers (Relative intensity of radiation) 24

2.2 Diffuse reflection of light 29

2.3 Diffuse transmission of light 35

2.4 The Radiation field in Deep Layers (Absolute Intensity) 41

2.5 The Atmospheric albedo for small true absorption 43

2.6 The Other Quantities in the case of small true absorption 46

Chapter 3 Atmospheres of Finite Optical Thickness 52

3.1 Diffuse reflection and transmission of light 52

3.2 Dependence of the reflections and transmission coefficients on optical thickness 57

3.3 Atmospheres of large optical thickness 60

3.4 Asymptotic formulas for auxiliary functions 65

3.5 Inhomogeneous atmospheres 66

Chapter 4 Atmospheres overlying a reflecting surface 74

4.1 Basic equations 74

4.2 The case of isotropic reflection 78

4.3 The albedo of the Atmosphere and Illumination of the surface 80

4.4 The spherical albedo of the planet 83

4.5 Specular reflection of light 86

Chapter 5 General Theory 89

5.1 Transformation of the basic integral equation 89

5.2 The Auxiliary equation 93

5.3 The function H^{n}(𝜂) 94

5.4 The fundamental function 𝛷^{m}(𝜏) 99

5.5 Particular cases 102

Chapter 6 General Theory (continued) 107

6.1 Expression of the source function in terms of auxiliary functions 107

6.2 The fundamental function 𝛷^{m}(𝜏, 𝜏_{0}) 109

6.3 112

6.4 Particular cases 115

6.5 Equations containing derivatives with respect to 𝜏_{0} 119

6.6 Atmospheres of large optical thickness 121

Chapter 7 Linear Integral equations for the reflection and transmission coefficients 126

7.1 Semi-infinite atmospheres 126

7.2 The radiation intensity averages over azimuth 131

7.3 Expressions in terms of the functions H^{n}(𝜂) 133

7.4 The case of three-term phase function 136

7.5 Numerical results 140

7.6 Atmospheres of finite optical thickness 143

7.7 Expressions in terms of the functions X^{m}(𝜂) and Y^{m}(𝜂) 147

7.8 The case of a two-term phase function 149

Chapter 8 Approximate Formulas 153

8.1 The use of integral relations 153

8.2 Some inequalities 156

8.3 Similarity relations 158

8.4 Directional averaging of the radiation intensity 161

8.5 The case of pure scattering 164

8.6 The Effect of the Reflection of Light by a Surface 167

8.7 The radiation field for Highly anisotropic scattering 169

Chapter 9 The radiation emerging from a planet 174

9.1 The distribution of brightness across a planetary disc 175

9.2 Dependance of planetary brightness on phase angle 177

9.3 Planetary spectra for different points on the disc 180

9.4 Planetary spectra for different phase angles 185

9.5 polarization of light from a planet 189

Chapter 10 Optical Properties of Planetary atmospheres 195

10.1 Interpretation of the photometric observations of Venus 195

10.2 Interpretation of polarimetric observations of Venus 198

10.3 The Atmosphere of the Earth 202

10.4 The Atmosphere of Mars 205

10.5 The Atmospheres of Giant Planets 207

Addendum 210

Chapter 11 Spherical Atmospheres 212

11.1 The Integral equation for the source function in the case of isotropic scattering 212

11.2 The basic equations of anisotropic scattering 218

11.3 Solution of the equation in particular case 221

11.4 The case of an absorption coefficient exponentially decreasing with altitude 224

11.5 Spacecraft observations of planets 231

Concluding remarks 235

Appendix 239

Author Index 250

Subject Index 253